Finite Group Databasebeta

← #13159503

Subgroups — #13159503

Order 1728.

860 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Dic3×C3⋊D3C211yesmaximal, characteristic
2#11722184C211yesmaximal, characteristic, complemented
3C3×#63647C211yesmaximal, characteristic
4C32×C8×Dic3C211yesmaximal, characteristic, complemented
5#11722050C211yesmaximal, characteristic, complemented
6#11722023C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10592384—31nomaximal
9#10590655—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×Dic3×C3⋊D3C411yescharacteristic
12C2×Dic3×C3⋊D3C411yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3×C3⋊Dic3C411yescharacteristic
20Dic3×C3⋊Dic3C411yescharacteristic
21C42×C3⋊D3—31no—
22C4×D3×Dic3—34no—
23C32×C4×C8—31no—
24#63647—31no—
25#63647D311yescharacteristic
26#63641—31no—
27#63641—31no—
28#63641—34no—
29#63641—64no—
30C4×C3⋊(C3⋊C8)—31no—
31C3×(C2×D3)⋊C8—34no—
32C3×C8×Dic3D314yescomplemented
33#63572—34no—
34(C2×D3)⋊(C3⋊C8)—34no—
35Dic3×C3⋊C8—34no—
36C22×C3×C3⋊D3C2×C411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
39C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
40C2×C3×C3⋊Dic3C2×C411yescharacteristic
41C3×C4×C3⋊D3—21no—
42C3×C4×C3⋊D3—21no—
43C2×C32×Dic3C2×C411yescharacteristic
44C2×C32×Dic3C2×C411yescharacteristic
45Dic3×C3⋊D3—22no—
46Dic3×C3⋊D3C814yescomplemented
47C33×C8—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C32×C3⋊C8—21no—
51#4718—91no—
52#4685—94no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×C3×C4×D3—34no—
57C2×D3×Dic3—34no—
58C2×D3×Dic3—34no—
59C2×C32×C8—24no—
60C2×C32×C8—31no—
61C2×C32×C8C2×D314yes—
62C2×C32×C8C2×D311yescharacteristic
63C32×C42—31no—
64C4×C3⋊Dic3—31no—
65C4×C3⋊Dic3—34no—
66C4×C3⋊Dic3—31no—
67C4×C3⋊Dic3—31no—
68C4×C3⋊Dic3—64no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4×Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic32—34no—
77Dic32—34no—
78C22×C33C22×C411yescharacteristic
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3C2×C812yes—
82C33×C4#4111yescharacteristic
83C33×C4#4111yescharacteristic
84C3⋊(C3⋊Dic3)—21no—
85C3⋊(C3⋊Dic3)C2×C812yes—
86C3×C3⋊Dic3—21no—
87C32×Dic3C2×C812yes—
88C32×Dic3M1612yes—
89C2×C4×Dic3—91no—
90C42×D3—94no—
91C3×C22⋊C8—91no—
92C3×C4×C8—34no—
93#830—91no—
94(C2×D3)⋊C8—34no—
95(C2×D3)⋊C8—94no—
96Dic3⋊C8—94no—
97Dic3⋊C8—94no—
98Dic3⋊C8—91no—
99Dic3⋊C8—91no—
100Dic3⋊C8—184no—
101C8×Dic3—91no—
102C8×Dic3C3⋊D311yescharacteristic, complemented
103C4×C3⋊C8—94no—
104C22×C3⋊D3C4×D311yescharacteristic
105C22×C3×D3—34no—
106C2×C32×C4—24no—
107C2×C32×C4—31no—
108C2×C32×C4—31no—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3—34no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3C4×D311yescharacteristic
118C4×C3⋊D3—21no—
119C4×C3⋊D3—21no—
120C4×C3⋊D3—34no—
121C4×C3⋊D3—62no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3C4×D314yes—
124C2×C3×Dic3C4×D314yes—
125C3×C4×D3—64no—
126C3×C4×D3—64no—
127D3×Dic3—316no—
128D3×Dic3—68no—
129C32×C8—21no—
130C32×C8—24no—
131C32×C8—44no—
132C32×C8—61no—
133C3⋊(C3⋊C8)—21no—
134C3⋊(C3⋊C8)—61no—
135C3⋊(C3⋊C8)—64no—
136C3⋊(C3⋊C8)—124no—
137C3×C3⋊C8—24no—
138C3×C3⋊C8—64no—
139#431—271noSylow
140C2×C33#11711yescharacteristic
141C2×C33C22×C811yescharacteristic
142C2×C33C2×M1611yescharacteristic
143C3×C3⋊D3—22no—
144C22×C3×C4—91no—
145C22×Dic3—91no—
146C22×Dic3—91no—
147C2×C4×D3—34no—
148C2×C4×D3—94no—
149C2×C4×D3—94no—
150C2×C3×C8—24no—
151C2×C3×C8—34no—
152C2×C3×C8—91no—
153C2×C3×C8D3214yes—
154C2×C3×C8C2×C3⋊D311yescharacteristic
155C3×C42—34no—
156C4×Dic3—94no—
157C4×Dic3—94no—
158C4×Dic3—94no—
159C4×Dic3—91no—
160C4×Dic3—91no—
161C4×Dic3—91no—
162C4×Dic3—184no—
163C4×Dic3C2×C3⋊D311yescharacteristic
164C2×C3⋊C8—34no—
165C2×C3⋊C8—94no—
166C2×C3⋊C8—91no—
167C2×C3⋊C8—184no—
168C2×C3⋊C8C2×C3⋊D311yescharacteristic
169C22×C32—24no—
170C22×C32C2×C4×D314yes—
171C22×C32C2×C4×D311yescharacteristic
172C2×C3⋊D3—21no—
173C2×C3⋊D3—21no—
174C2×C3⋊D3D3×C812yes—
175C2×C3×D3—38no—
176C2×C3×D3—64no—
177C2×C3×D3—64no—
178C32×C4—24no—
179C32×C4—24no—
180C32×C4—32no—
181C32×C4—32no—
182C32×C4#23514yes—
183C32×C4#23514yes—
184C32×C4#23711yescharacteristic
185C32×C4#23711yescharacteristic
186C3⋊Dic3—21no—
187C3⋊Dic3—32no—
188C3⋊Dic3—38no—
189C3⋊Dic3—68no—
190C3⋊Dic3—61no—
191C3⋊Dic3—64no—
192C3⋊Dic3—124no—
193C3×Dic3—64no—
194C3×Dic3D3×C818yes—
195C3×Dic3ΓL(1,𝔽25)18yes—
196C2×C42—271no—
197C4⋊C8—271no—
198C4⋊C8—271no—
199C22⋊C8—91no—
200C22⋊C8—271no—
201C4×C8—31no—
202C4×C8—271no—
203C33#43111yescharacteristic, complemented, Sylow
204C23×C3—91no—
205C22×D3—34no—
206C2×C3×C4—24no—
207C2×C3×C4—34no—
208C2×C3×C4—34no—
209C2×C3×C4—91no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4C2×D3214yes—
213C2×C3×C4C22×C3⋊D311yescharacteristic
214C2×Dic3—34no—
215C2×Dic3—94no—
216C2×Dic3—91no—
217C2×Dic3—94no—
218C2×Dic3—94no—
219C2×Dic3—91no—
220C2×Dic3—184no—
221C2×Dic3—184no—
222C2×Dic3—182no—
223C2×Dic3C4×C3⋊D311yescharacteristic
224C2×Dic3C4×C3⋊D311yescharacteristic
225C4×D3—64no—
226C4×D3—64no—
227C4×D3—916no—
228C4×D3—188no—
229C3×C8—24no—
230C3×C8—21no—
231C3×C8—44no—
232C3×C8—64no—
233C3×C8—181no—
234C3⋊C8—21no—
235C3⋊C8—64no—
236C3⋊C8—184no—
237C3⋊C8—181no—
238C3⋊C8—364no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32#87214yes—
243C2×C32#87711yescharacteristic
244C2×C32C2×D3×C814yes—
245C2×C32C2×D3×C811yescharacteristic
246C2×C32C2×ΓL(1,𝔽25)14yes—
247C2×C32D3×M1611yescharacteristic
248C3⋊D3—22no—
249C3×D3—68no—
250C22×C4—91no—
251C22×C4—271no—
252C22×C4—271no—
253C2×C8—31no—
254C2×C8—91no—
255C2×C8—271no—
256C2×C8D3×C3⋊D311yescharacteristic
257C42—31no—
258C42—271no—
259C42—271no—
260C42—271no—
261C22×C3—24no—
262C22×C3—92no—
263C22×C3—181no—
264C22×C3—181no—
265C22×C3C4×D3214yes—
266C22×C3C2×C4×C3⋊D311yescharacteristic
267C2×D3—38no—
268C2×D3—64no—
269C2×D3—64no—
270C3×C4—24no—
271C3×C4—24no—
272C3×C4—38no—
273C3×C4—38no—
274C3×C4—181no—
275C3×C4#379514yes—
276C3×C4#380414yes—
277C3×C4#382811yescharacteristic
278C3×C4#382811yescharacteristic
279Dic3—64no—
280Dic3—98no—
281Dic3—92no—
282Dic3—188no—
283Dic3—184no—
284Dic3—181no—
285Dic3—364no—
286Dic3C8×C3⋊D312yes—
287Dic3#374312yes—
288C32—24no—
289C32#468514yescomplemented
290C32#471811yescharacteristic, complemented
291C23—91no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—91no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—274no—
298C2×C4—271no—
299C2×C4—271no—
300C2×C4—542no—
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—21no—
303C8—61no—
304C8—181no—
305C8—541no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—182no—
310C2×C3D32×C814yes—
311C2×C3D3×ΓL(1,𝔽25)14yes—
312C2×C3#6387714yes—
313C2×C3#6408111yescharacteristic
314C2×C3C2×C8×C3⋊D311yescharacteristic
315C2×C3C2×#374311yescharacteristic
316D3—68no—
317C22—92no—
318C22—181no—
319C22—181no—
320C22C4×D3×C3⋊D311yescharacteristic
321C4—32no—
322C4—32no—
323C4—181no—
324C4—272no—
325C4—541no—
326C4#8902111yescharacteristic
327C4#8903111yescharacteristic
328C3—24no—
329C3#1059065514yescomplemented
330C3#1059238411yescharacteristic, complemented
331C2—182no—
332C2D3×C8×C3⋊D311yescharacteristic
333C2D3×#374311yescharacteristic
334C2#1172346411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutatorC2×C33C22×C8
FittingC2×C33×C8C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#431
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11722184C2semidirect
4. C32×C8×Dic3C2semidirect
5. #11722050C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
32. C3×C8×Dic3D3semidirect
46. Dic3×C3⋊D3C8semidirect
102. C8×Dic3C3⋊D3semidirect
203. C33#431semidirect
289. C32#4685semidirect
290. C32#4718semidirect
329. C3#10590655semidirect
330. C3#10592384semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×Dic3×C3⋊D32
#117221842
C3×#636472
C32×C8×Dic32
#117220502
#117220232
Dic3×C3⋊(C3⋊C8)2
#105923843
#105906553